(G) Sum to Infinity of Geometric Progressions (Part 1)

Posted on April 22, 2020 by user

1.4.4 Sum to Infinity of Geometric Progressions

(G) Sum to Infinity of Geometric Progressions

$S \infty = \frac{a}{1 - r}, - 1 < r < 1$

a = first term

r = common ratio

S∞ = sum to infinity

Example:

Find the sum to infinity of each of the following geometric progressions.

(a) 8, 4, 2, ...

(b)

$\frac{2}{3}, \frac{2}{9}, \frac{2}{27}, .....$

(c) 3, 1, ⅓, ….

Solution:

(a)

8, 4, 2, ….

a = 2, r = 4/8 = ½
S∞ = 8 + 4 + 2 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + …..

$S \infty = \frac{a}{1 - r} = \frac{2}{1 - \frac{1}{2}} = 4$

(b)

$\begin{array}{l} \frac{2}{3}, \frac{2}{9}, \frac{2}{27}, ..... \\ a = \frac{2}{3}, r = \frac{2 / 9}{2 / 3} = \frac{1}{3} \\ S \infty = \frac{a}{1 - r} \\ S \infty = \frac{\frac{2}{3}}{1 - \frac{1}{3}} = 1 \end{array}$

(c)

$\begin{array}{l} 3, 1, \frac{1}{3}, ..... \\ a = 3, r = \frac{1}{3} \\ S \infty = \frac{a}{1 - r} \\ S \infty = \frac{3}{1 - \frac{1}{3}} = \frac{3}{2 / 3} = \frac{9}{2} \end{array}$